Parallel Curves

1954 ◽  
Vol 6 ◽  
pp. 99-107 ◽  
Author(s):  
G. P. Henderson
Keyword(s):  
Euclidean Space ◽  
Euclidean Plane ◽  
Parameter Family ◽  
Dimensional Euclidean Space

In the Euclidean plane a curve C has a one-parameter family of parallel involutes and a unique evolute C* which coincides with the locus of the centres of the osculating circles of C. If is parallel to C, C* is also the evolute of . We will study parallel curves in n-dimensional Euclidean space and obtain generalizations of the properties given above.We will study parallel curves in n-dimensional Euclidean space and obtain generalizations of the properties given above.

Moments of coverage and coverage spaces

1966 ◽  
Vol 3 (02) ◽  
pp. 550-555 ◽  
Author(s):  
Gedalia Ailam
Keyword(s):  
Distribution Function ◽  
Euclidean Space ◽  
Euclidean Plane ◽  
Random Sets ◽  
Dimensional Euclidean Space ◽  
Homogeneous Distribution ◽  
Transformation Formulas

Moments of the measure of the intersection of a given set with the union of random sets may be called moments of coverage. These moments specified for the case of independent random sets in the Euclidean space, were treated by several authors. Kolmogorov (1933) and Robbins (1944), (1945), (1947) derived transformation formulas for the moments of coverage under some specified conditions. Robbins' formula was applied by Robbins (1945), Santaló (1947) and Garwood (1947) to computations of expectations and variances of coverage in the cases of spheres and intervals in 2-dimensional and in n-dimensional Euclidean space. The computations were carried out under the assumption of a simple distribution function for the centers of the covering figures (a homogeneous distribution in most of the cases and a normal one in some of them). Neyman and Bronowski (1945) computed by a different method the variance of coverage for some specific cases in the Euclidean plane.

Moments of coverage and coverage spaces

1966 ◽  
Vol 3 (2) ◽  
pp. 550-555 ◽  
Author(s):  
Gedalia Ailam
Keyword(s):  
Distribution Function ◽  
Euclidean Space ◽  
Euclidean Plane ◽  
Random Sets ◽  
Dimensional Euclidean Space ◽  
Homogeneous Distribution ◽  
Transformation Formulas

Moments of the measure of the intersection of a given set with the union of random sets may be called moments of coverage. These moments specified for the case of independent random sets in the Euclidean space, were treated by several authors. Kolmogorov (1933) and Robbins (1944), (1945), (1947) derived transformation formulas for the moments of coverage under some specified conditions. Robbins' formula was applied by Robbins (1945), Santaló (1947) and Garwood (1947) to computations of expectations and variances of coverage in the cases of spheres and intervals in 2-dimensional and in n-dimensional Euclidean space. The computations were carried out under the assumption of a simple distribution function for the centers of the covering figures (a homogeneous distribution in most of the cases and a normal one in some of them). Neyman and Bronowski (1945) computed by a different method the variance of coverage for some specific cases in the Euclidean plane.

scholarly journals Implicit Equations of the Henneberg-Type Minimal Surface in the Four-Dimensional Euclidean Space

Mathematics ◽  
2018 ◽  
Vol 6 (12) ◽  
pp. 279
Author(s):  
Erhan Güler ◽  
Ömer Kişi ◽  
Christos Konaxis
Keyword(s):  
Euclidean Space ◽  
Minimal Surface ◽  
Parameter Family ◽  
Weierstrass Representation ◽  
Dimensional Euclidean Space ◽  
Algebraic Equations ◽  
Implicit Equations ◽  
Positive Integers ◽  
Two Parameter

Considering the Weierstrass data as ( ψ , f , g ) = ( 2 , 1 - z - m , z n ) , we introduce a two-parameter family of Henneberg-type minimal surface that we call H m , n for positive integers ( m , n ) by using the Weierstrass representation in the four-dimensional Euclidean space E 4 . We define H m , n in ( r , θ ) coordinates for positive integers ( m , n ) with m ≠ 1 , n ≠ - 1 , - m + n ≠ - 1 , and also in ( u , v ) coordinates, and then we obtain implicit algebraic equations of the Henneberg-type minimal surface of values ( 4 , 2 ) .

scholarly journals Multiple Lattice Tilings in Euclidean Spaces

2018 ◽  
Vol 62 (4) ◽  
pp. 923-929
Author(s):  
Qi Yang ◽  
Chuanming Zong
Keyword(s):  
Euclidean Space ◽  
Convex Domain ◽  
Euclidean Plane ◽  
Dimensional Euclidean Space ◽  
Euclidean Spaces ◽  
Convex Domains ◽  
Lattice Tiling ◽  
Centrally Symmetric

AbstractIn 1885, Fedorov discovered that a convex domain can form a lattice tiling of the Euclidean plane if and only if it is a parallelogram or a centrally symmetric hexagon. This paper proves the following results. Except for parallelograms and centrally symmetric hexagons, there are no other convex domains that can form two-, three- or four-fold lattice tilings in the Euclidean plane. However, there are both octagons and decagons that can form five-fold lattice tilings. Whenever $n\geqslant 3$, there are non-parallelohedral polytopes that can form five-fold lattice tilings in the $n$-dimensional Euclidean space.

ON ENUMERATING AND SELECTING DISTANCES

2001 ◽  
Vol 11 (03) ◽  
pp. 291-304 ◽  
Author(s):  
TIMOTHY M. CHAN
Keyword(s):  
Euclidean Space ◽  
Euclidean Plane ◽  
Randomized Algorithm ◽  
Dimensional Euclidean Space ◽  
Running Time ◽  
Selection Problems ◽  
Deterministic Algorithms ◽  
Parametric Search ◽  
Enumeration Problem ◽  
Point Set

Given an n-point set, the problems of enumerating the k closest pairs and selecting the k-th smallest distance are revisited. For the enumeration problem, we give simpler randomized and deterministic algorithms with O(n log n+k) running time in any fixed-dimensional Euclidean space. For the selection problem, we give a randomized algorithm with running time O(n log n+n2/3k1/3 log 5/3n) in the Euclidean plane. We also describe output-sensitive results for halfspace range counting that are of use in more general distance selection problems. None of our algorithms requires parametric search.

scholarly journals Symmetries of Monocoronal Tilings

2015 ◽  
Vol Vol. 17 no.2 (Combinatorics) ◽  
Author(s):  
Dirk Frettlöh ◽  
Alexey Garber
Keyword(s):  
Euclidean Space ◽  
Euclidean Plane ◽  
Dimensional Euclidean Space ◽  
Symmetry Groups ◽  
Translation Group ◽  
One Dimensional ◽  
Higher Dimensional ◽  
International Audience

International audience The vertex corona of a vertex of some tiling is the vertex together with the adjacent tiles. A tiling where all vertex coronae are congruent is called monocoronal. We provide a classification of monocoronal tilings in the Euclidean plane and derive a list of all possible symmetry groups of monocoronal tilings. In particular, any monocoronal tiling with respect to direct congruence is crystallographic, whereas any monocoronal tiling with respect to congruence (reflections allowed) is either crystallographic or it has a one-dimensional translation group. Furthermore, bounds on the number of the dimensions of the translation group of monocoronal tilings in higher dimensional Euclidean space are obtained.

On Vitali's Theorem for Groups of Motions of Euclidean Space

1999 ◽  
Vol 6 (4) ◽  
pp. 323-334
Author(s):  
A. Kharazishvili
Keyword(s):  
Euclidean Space ◽  
Dimensional Euclidean Space ◽  
Finite Dimensional

Abstract We give a characterization of all those groups of isometric transformations of a finite-dimensional Euclidean space, for which an analogue of the classical Vitali theorem [Sul problema della misura dei gruppi di punti di una retta, 1905] holds true. This characterization is formulated in purely geometrical terms.

Some properties of Wigner coefficients and hyperspherical harmonics

1956 ◽  
Vol 52 (3) ◽  
pp. 424-430 ◽  
Author(s):  
A. P. Stone
Keyword(s):  
Angular Momentum ◽  
Euclidean Space ◽  
Rotation Group ◽  
Dimensional Euclidean Space ◽  
Hyperspherical Harmonics ◽  
Shift Operators ◽  
Analytical Relations

ABSTRACTGeneral shift operators for angular momentum are obtained and applied to find closed expressions for some Wigner coefficients occurring in a transformation between two equivalent representations of the four-dimensional rotation group. The transformation gives rise to analytical relations between hyperspherical harmonics in a four-dimensional Euclidean space.

On inequalities of the Tchebychev type

1963 ◽  
Vol 59 (1) ◽  
pp. 135-146 ◽  
Author(s):  
J. F. C. Kingman
Keyword(s):  
Probability Theory ◽  
Euclidean Space ◽  
Random Variable ◽  
Dimensional Euclidean Space ◽  
Real Random Variable ◽  
Finite Dimensional

1. A type of problem which frequently occurs in probability theory and statistics can be formulated in the following way. We are given real-valued functions f(x), gi(x) (i = 1, 2, …, k) on a space (typically finite-dimensional Euclidean space). Then the problem is to set bounds for Ef(X), where X is a random variable taking values in , about which all we know is the values of Egi(X). For example, we might wish to set bounds for P(X > a), where X is a real random variable with some of its moments given.

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